Lattice (order) - Examples

Examples

• For any set A, the collection of all subsets of A (called the power set of A) can be ordered via subset inclusion to obtain a lattice bounded by A itself and the null set. Set intersection and union interpret meet and join, respectively.
• For any set A, the collection of all finite subsets of A, ordered by inclusion, is also a lattice, and will be bounded if and only if A is finite.
• For any set A, the collection of all partitions of A, ordered by refinement, is a lattice.
• The natural numbers (including 0) in their usual order form a lattice, under the operations of "min" and "max". 0 is bottom; there is no top.
• The Cartesian square of the natural numbers, ordered by ≤ so that (a,b) ≤ (c,d) ↔ (a ≤ c) & (b ≤ d). (0,0) is bottom; there is no top.
• The natural numbers also form a lattice under the operations of taking the greatest common divisor and least common multiple, with divisibility as the order relation: a ≤ b if a divides b. 1 is bottom; 0 is top.
• Any complete lattice (also see below) is a (rather specific) bounded lattice. This class gives rise to a broad range of practical examples.
• The set of compact elements of an arithmetic complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice. This is the specific property which distinguishes arithmetic lattices from algebraic lattices, for which the compacts do only form a join-semilattice. Both of these classes of complete lattices are studied in domain theory.

Most posets are not lattices, including the following.

• A discrete poset, meaning a poset such that x ≤ y implies x = y, is a lattice if and only if it has at most one element. In particular the two-element discrete poset is not a lattice.
• Although the set {1,2,3,6} partially ordered by divisibility is a lattice, the set {1,2,3} so ordered is not a lattice because the pair 2,3 lacks a join, and it lacks a meet in {2,3,6}.
• The set {1,2,3,12,18,36} partially ordered by divisibility is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2,3 has three upper bounds, namely 12, 18, and 36, none of which is the least of those three under divisibility (12 and 18 do not divide each other). Likewise the pair 12,18 has three lower bounds, namely 1, 2, and 3, none of which is the greatest of those three under divisibility (2 and 3 do not divide each other).

Further examples of lattices are given for each of the additional properties discussed below.